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Mathematics 2019, 7(2), 119; https://doi.org/10.3390/math7020119

Modified Relaxed CQ Iterative Algorithms for the Split Feasibility Problem

1
College of Air Traffic Management, Civil Aviation University of China, Tianjin 300300, China
2
College of Science, Civil Aviation University of China, Tianjin 300300, China
*
Author to whom correspondence should be addressed.
Supported by National Natural Science Foundation of China (No. 61571441) and Scientic Research project of Tianjin Municipal Education Commission (No. 2018KJ253).
Received: 20 November 2018 / Revised: 17 January 2019 / Accepted: 21 January 2019 / Published: 23 January 2019
(This article belongs to the Special Issue Fixed Point Theory and Related Nonlinear Problems with Applications)
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Abstract

The split feasibility problem models inverse problems arising from phase retrievals problems and intensity-modulated radiation therapy. For solving the split feasibility problem, Xu proposed a relaxed CQ algorithm that only involves projections onto half-spaces. In this paper, we use the dual variable to propose a new relaxed CQ iterative algorithm that generalizes Xu’s relaxed CQ algorithm in real Hilbert spaces. By using projections onto half-spaces instead of those onto closed convex sets, the proposed algorithm is implementable. Moreover, we present modified relaxed CQ algorithm with viscosity approximation method. Under suitable conditions, global weak and strong convergence of the proposed algorithms are proved. Some numerical experiments are also presented to illustrate the effectiveness of the proposed algorithms. Our results improve and extend the corresponding results of Xu and some others. View Full-Text
Keywords: split feasibility problem; relaxed CQ algorithm; convergence; Hilbert space split feasibility problem; relaxed CQ algorithm; convergence; Hilbert space
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Wang, X.; Zhao, J.; Hou, D. Modified Relaxed CQ Iterative Algorithms for the Split Feasibility Problem. Mathematics 2019, 7, 119.

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